In the log geometric mirror construction, one can show that after fixing a point constraint, the tropical type above defines a moduli space of virtual dimension 0, and the structure constants defining \(R_{(X,D)}\)  are the degrees of the relevant virtual fundamental class. In the non-archimedean mirror construction, a well behaved moduli space is not given directly from this tropical data, but only after extending the bounded leg to an extended leg. This extension is required to be straight in the affine structure put on \(B\) , given via an identification \(B \cong M_\mathbb {R}\)  which exists due to the choice of Zariski dense \(n\)  dimensional torus sitting inside \(U\) , see Lemma \(2.2\)  of [1]}. Keel and Yu show these curves vary in connected families in the relevant ambient analytic moduli space of rational curves, and after picking a point in \(p \in B \subset U^{an}\) , there are a finite number of curves containing the point which have no automorphisms, and the structure constants are given by the degree of this 0-dimensional reduced scheme, or a simple count of curves. No virtual class is needed in this construction, as the relevant moduli space locally around contributing curves turns out to be smooth of the expected dimension, as established in Section 3 of [1]}. Using these respective definitions, the main results of [1]} and [4]} establish that these structure constants give \(R_{(X,D)}\)  the structure of an \(S_X\) -algebra.
